Defining an equivalence relation

An equivalence relation exists if it meet the following three properties, being reflexive, symmetric, and transitive.

We have a set A and R is a relation on that set.

1. R is reflexive on A if and only if for all x that is an element of A, x relates to x, or x R x.

2. R is symmetric if and only if for all x and y in A the relation of x to y is the same as the relation of y to x.

3. R is transitive if and only if for all x,y, and z that are elements in A, if x relates to y, and y relates to z, then x relates to z.